Using the fancyhdr package, how do I have a footer but no header on the first page only

You might want to leave the plain style as it is and define a new style, let's say firstpage

\fancypagestyle{firstpage}{%
  \fancyhf{}%
  \renewcommand{\footrulewidth}{0.1mm}%
  \fancyfoot[R]{Assignment No.}%
  \fancyfoot[C]{\thepage}%
  \fancyfoot[L]{\today}%
  \renewcommand{\headrulewidth}{0mm}%
} 

and, at the very beginning of the document, issue

\thispagestyle{firstpage}

MWE:

\documentclass{article}

\usepackage{fancyhdr}

\fancypagestyle{firstpage}{%
  \fancyhf{}%
  \renewcommand{\footrulewidth}{0.1mm}%
  \fancyfoot[R]{Assignment No.}%
  \fancyfoot[C]{\thepage}%
  \fancyfoot[L]{\today}%
  \renewcommand{\headrulewidth}{0mm}%
}  

\fancyhf{}
\renewcommand{\footrulewidth}{0.1mm}
\fancyfoot[R]{Assignment No.}
\fancyfoot[C]{\thepage}
\fancyfoot[L]{\today}
\fancyhead[L]{\leftmark}
\fancyhead[R]{Name:First Last}

\pagestyle{fancy}

\usepackage{blindtext} % just for the example

\begin{document}

\thispagestyle{firstpage}

\blinddocument

\end{document} 

Only the very first page has no header:

\documentclass{book}    
\usepackage{fancyhdr}   
\pagestyle{fancy}
\fancyhf{}
\renewcommand{\footrulewidth}{0.1mm}
\fancyfoot[R]{Assignment No.}
\fancyfoot[C]{\thepage}
\fancyfoot[L]{\today}
\fancyhead[L]{\leftmark}
\fancyhead[R]{Name:First Last}

\fancypagestyle{plain}{%
  \fancyhf{}%
  \renewcommand{\footrulewidth}{0.1mm}%
  \fancyfoot[R]{Assignment No.}%
  \fancyfoot[C]{\thepage}%
  \fancyfoot[L]{\today}%
  \renewcommand{\headrulewidth}{0mm}%
}   
\usepackage{blindtext}

\begin{document}

\blinddocument
\makeatletter\let\ps@plain\ps@fancy\makeatother % anywhere after the first page/chapter

\blinddocument

\end{document}

According to the fancyhdr manual the \fancypagestyle command is what you're looking for (see section 7 of the manual):

This command can be used to redefine existing pagestyles (like plain) or to define new ones, e.g. if part of your document is to use a different pagestyle.

\documentclass{scrartcl}
\usepackage{fancyhdr}

\pagestyle{fancy}
\fancypagestyle{fancy}{%
\fancyhf{}
\renewcommand{\footrulewidth}{0.1mm}
\fancyfoot[R]{Assignment No.}
\fancyfoot[C]{\thepage}
\fancyfoot[L]{\today}
\fancyhead[L]{\leftmark}
\fancyhead[R]{Name:First Last}}


\fancypagestyle{plain}{%
\fancyhf{} % clear all header and footer fields
\fancyfoot[R]{Assignment No.}
\fancyfoot[C]{\thepage}
\fancyfoot[L]{\today}
\renewcommand{\footrulewidth}{0.1mm}}

\usepackage{blindtext}
\title{Test document}
\begin{document}
\maketitle

\blinddocument

\end{document}